Gate-induced decoupling of surface and bulk state properties in selectively-deposited Bi$_2$Te$_3$ nanoribbons

Thank you very much for your honest feedback and the fruitful discussions. We recognized your concerns and were glad to realize the inconsistency in our data analysis, based on your comments. We also recognized our confusion based on the previous reviews and our response thereon. A detailed discussion of your response will follow in the point-to-point response below.

The Referee writes: Despite the improvements made by the authors in the revised version of the manuscript, I am still not convinced that this work meets the requirements of SciPost. Even if the ability to reproduce such challenging measurements in systems that differ from previously investigated devices can be consider as an important step toward the understanding of the transport properties of 3DTIs nanowire, I do not agree that the present work brings significant novelty to the field. I still believe that some interpretations that are essential for the message delivered by the authors are still rather confusing to the reader. I would like again to insist that this is not a statement about the quality of the work presented in the paper that is obviously a solid work with reliable results.

Our response: Our analysis is an important benchmarking step for 3D TI nanoribbon grown using our novel deposition technique and needed some adaptations to existing models. With respect to the novelty of our approach we acknowledge that there is indeed overlap with previous work. We would agree to a transfer to SciPost Physics Core, when inconsistencies in our data analysis are clarified and both referees as well as the editor in charge agree as well.

The Referee writes: I comment below these two points in more details: Novelty requirements: - The authors mention in their answer and in the manuscript that one of the novelties is the use of MBE growth that allow to have a fabrication process compatible with standard CMOS processing. Nevertheless, despite a strong interest that such growth methods raise in the community, the technique has been already presented in several previous publications of the same group. Moreover, the paper does not focus on the growth properties but rather on the transport properties of the nanowires and on the analysis of the transport results. As already mentioned earlier in the first report, such results have been already reported in the literature as cited in the manuscript.

Our response: The analysis we demonstrate in this publication is indeed in line with a few of our publications on selectively defined 3D TI structures/devices. Our own referenced work, however, demonstrates different analysis aspects of these structures, which should be in favor for both this as well as our previous publications. Our results benchmark the quality of our approach and fuels our ongoing research.

The Referee writes: - A second important point concerns the analysis. Going carefully through the Ref. 18 (PRB 97, 035157), the model developed by Ziegler et al. is very similar than the one presented here. For electrostatic simulations, Ziegler et al. indeed consider a nanowire made of a perfect metal that entirely screen the electric field that vanishes hence completely in the nanowire. They attribute the origin of the entire “capacitive” charge to the surface states. Thus, the only assumption that they did is that surface states perfectly screen the electric field. Such a screening induces a shift of the Fermi energy that strongly depends on the circumferential coordinate “s”. In the absence of further clarifications, I am not able to catch the essence of the model extension Rosenbach et al. claim to do. Did they introduce any quantum capacitance effect to account for the screening by both bulk (perfect) metallic states and (finite density of states) TSS? In other term, how n$_{2D}^{TSS}$ (s) is determined is not clear to me and the explanation of the model is confused. The results obtained from the simulation are moreover very similar to what Ziegler et al. show in their work.

Our response: We indeed adhere to the analysis described by Ziegler et al. before. We agree with the assumption that as soon as there are coherent states along the perimeter of 3D TI nanoribbons an effective capacitance need to be considered in otherwise asymmetric gating profiles using only a top gate electrode. The extension we performed are on the determination of the effective capacitance as carefully described in section 2.2. 'Field effect in TI ribbon devices'. This adaptation has been necessary for our analysis as our data shows bulk contributions throughout the whole range of gate voltages applied as well as due to a different Fermi level/ different intrinsic charge carrier density on the top and bottom surface of our nanoribbon devices. The n$_{2D}^{TSS}$ (s) data is not determined from our model but from experimental work presented before. The values of the intrinsic charge carrier densities are based on the analysis of Shubnikov-de Haas data presented in:

D. Rosenbach et al., 'Quantum Transport in Topological Surface States of Selectively Grown Bi2Te3 Nanoribbons', Adv. El. Mat. 6(8), 2000205 (2020), doi:10.1002/aelm.202000205. (Reference 22 in the manuscript)

as well as results from ARPES measurements performed by our close collaborators:

M. Eschbach et al., 'Bi1Te1 is a dual topological insulator', Nat. Commun. 8(1), 14976 (2017), doi:10.1038/ncomms14976. (Reference 33 in the manuscript) L. Plucinski et al., 'Robust surface electronic properties of topological insulators: Bi2te3 films grown by molecular beam epitaxy', Appl. Phys. Let. 98(22), 222503 (2011), doi:10.1063/1.3595309. (Reference 34 in the manuscript)

This is as well described in the second paragraph of section 3.1.

The Referee writes: Interpretation in terms of independent 1D subbands: The distinction between the formation of 1D subbands and the observation of AB oscillations is puzzling in the manuscript as well as in the overall answer of the authors. If both can be related together, they can also have different origins. According to my understanding, there is two relevant length scales to consider for confinement effects and for the observation of AB oscillations: the mean free path $l_0$ and the phase coherence length $l_{\phi}$, with generally $l_0$ $<$ $l_{\phi}$. Three different regimes can be distinguished, considering the value of perimeter of the nanowire P:

(i) $P < l_0 < l_{\phi}$ : this regime corresponds to quantum confinement where the band structure can be seen as a collection of 1D subbands as described in the article. In that case, AB oscillations can be observed and are related to the 1D subbands: they have the same origin. Their amplitude is typically given by PRB 97, 075401 as mentioned by the authors. In this case, the interpretation in the framework of the diffusive regime cannot be done.

(ii) $l_0 < P < l_{\phi}$ : in this regime the “k” vector is not a good quantum number anymore the description in terms of independent 1D subbands does not hold anymore. In other terms, the subbands are strongly coupled to each other and the band structure is very close to the one of Dirac cones as seen in TSS without any confinement. Since P $<$ l$_{phi}$, AB oscillations can be nevertheless measured and should be analyzed in the framework of diffusive transport are the authors did in the present work.

(iii) $l_0 < l_{\phi} < P $: No observation of neither AB nor 1D subbands should be possible. In the manuscript, the authors go along with the scenario (i) (despite the fact that they clearly state to have $l_0$ $<$ P) and latter with the scenario (ii). This has some particular important consequences for the analysis of $G(V_g)$ and for the attribution to the fluctuations to 1D subbands effects rather than for quantum interferences effect. If this assumption was justified for Ziegler et al. for instance who use to deal with high mobility HgTe thin films, it cannot be the case here.

Our response: Thank you for this detailed description, this was indeed helpful to re-evaluate our analysis. Based on the analysis shown in Ziegler et al. the combination of actually showing full-integer flux quantum periodic oscillations and the subsequent identification of a spin degeneracy factor of $g_s=1$ in the electrostatic modeling is a strong evidence of topologically non-trivial states being responsible for the observation of periodic oscillations in the magnetoconductance data. With respect to the line of arguments presented in Ziegler et al. we made sure to include proper arguments within the second and third paragraph of section 3.2 of the manuscript.

The main inconsistency we therefore see in the analysis of the phase-coherence length from our temperature dependent AB oscillation amplitude analysis. If we agree that above mentioned argument concludes the origin of observed oscillations are topologically non-trivial states, this excludes the use of a diffusive model for our data. This is what has already been pointed out in the previous round of discussion. We agree that both descriptions (electrostatics as well as temperature dependency) of the same oscillations would need to be based on the same dimensionality and mode of transport to be considered. Following previous works on quasi-1D systems (e.g. PRB, 64, 045327 (2011) and PRL 110, 186806 (2013)) the temperature dependent AB oscillation amplitude follows an exponential decay function, in which $l_{\phi}$ has been determined to be linearly dependent on the inverse of T. A fit under this assumption does indeed fit our data better than previously assumed, classical 2D diffusive system. The fit also results in a much greater phase coherence length than previously identified of around 1.7$\mu$m at 1K. As this value is larger than the circumference of the nanoribbon device, our assumption of ballistic surface transport is justified.

One additional remark to the temperature dependency of the AB oscillations in Fig. 4 b). Going through the data again we realized that we missed a factor of two in the displayed oscillation amplitude. We changed the data, which however does not alter the qualitative results of the fits performed.

The Referee writes: One last and less important remark about the screening length: the formula mentioned by the authors in their answer concerns the regime of a Fermi wavelength smaller than the screening length, a condition that is obviously not satisfied here. For a screening length smaller than the Fermi wavelength, the Thomas-Fermi wavelength should apply, leading to substantially longer screening length. Nevertheless, this should not change the conclusion given by the authors.

Our response: Thank you for your remark, we acknowledge our mistake but agree that this will not change our conclusions.

Thank you very much for the kind words and honest feedback on our manuscript. We indeed see the strength in our approach, as you nicely summarized, in benchmarking our novel growth technique by showing signatures of a magnetic field dependent dispersion of the topological surface states wrapped around our nanoribbon structures. As mentioned before this validates our approach suitable for more complex devices utilizing 3D TI nanoribbons.

We agree that our manuscript might not be suited for a publication in SciPost Physics and would agree to a transfer to SciPost Physics Core. However, the inconsistency in our data analysis that the other anonymous referee pointed out has to be resolved first and the editor in charge needs to agree that we meet the criteria for SciPost Physcis Core. For more details on these issues please refer to our response to the other referee and the resubmission letter, respectively.